Multiscale Finite Element Method for Non-Stationary Stokes-Darcy Model

Core Concepts

Combining multiscale finite element method for solving non-stationary Stokes-Darcy model efficiently.

Abstract

The paper introduces an algorithm that combines the multiscale finite element method to solve the non-stationary Stokes-Darcy model. It focuses on reducing computational costs by solving problems on relatively coarse grids while maintaining high accuracy. The algorithm involves parallel computation of multiscale basis functions in the Darcy region and employs an implicit-explicit scheme to solve the equations. By considering periodic permeability coefficients, stability and convergence of the algorithm are demonstrated. Three numerical experiments verify the effectiveness and accuracy of the proposed method.

Stats

Permeability coefficient exhibits multiscale characteristics.
Higher accuracy compared to standard finite element method.
Stability and convergence of the algorithm demonstrated through numerical experiments.

Quotes

"One significant feature of the algorithm is that it solves problems on relatively coarse grids, thus significantly reducing computational costs."
"Under the assumption that the permeability coefficient is periodic and independent of time, this paper demonstrates the stability and convergence of the algorithm."
"The rationality and effectiveness of the algorithm are verified through three numerical experiments, with experimental results consistent with theoretical analysis."

Key Insights Distilled From

Finite element method coupled with multiscale finite element method for the non-stationary Stokes-Darcy model

by Yachen Hong,... at arxiv.org 03-19-2024

https://arxiv.org/pdf/2403.11600.pdf

Deeper Inquiries

How does incorporating multiscale characteristics impact computational efficiency beyond cost reduction

Incorporating multiscale characteristics in algorithms can significantly improve computational efficiency by reducing the memory and CPU time required for solving complex problems. By using multiscale basis functions, the algorithm can capture information at different scales simultaneously, leading to more accurate results on coarser grids. This not only reduces computational costs but also enhances the overall efficiency of the solution process. Additionally, by focusing on macroscopic features rather than microscopic details, the algorithm can provide a more streamlined approach to problem-solving, further enhancing computational efficiency.

What potential limitations or challenges could arise when applying this algorithm to more complex systems

When applying this algorithm to more complex systems, several limitations and challenges may arise. One challenge is related to the accuracy of the results obtained on coarse grids. While multiscale methods aim to capture fine-scale features on coarser meshes, there may still be some loss of detail compared to high-resolution simulations. Another limitation is scalability; as the complexity of the system increases, so does the computational demand, potentially leading to resource constraints.
Furthermore, ensuring stability and convergence in highly nonlinear or heterogeneous systems can be challenging with multiscale methods. The choice of appropriate basis functions and mesh sizes becomes crucial in maintaining accuracy while managing computational costs effectively. Lastly, implementing boundary conditions and interface treatments accurately in complex geometries poses another challenge that needs careful consideration during algorithm development.

How can this research contribute to advancements in other fields beyond fluid dynamics

The research presented here on combining finite element methods with multiscale techniques for solving coupled Stokes-Darcy models has broader implications across various fields beyond fluid dynamics:

Materials Science: The methodology developed for capturing multiscale phenomena can be applied in materials science for studying composite materials with heterogeneous properties.

Biomedical Engineering: In biomedical engineering applications such as modeling blood flow through tissues or drug delivery mechanisms within porous media structures.

Environmental Engineering: Understanding groundwater contamination transport through porous media or simulating pollutant dispersion in natural aquifers.

Renewable Energy: Analyzing heat transfer processes in geothermal reservoirs or optimizing energy extraction from subsurface resources like oil wells.

By advancing numerical techniques that account for multiscale behaviors efficiently and accurately, this research opens up possibilities for innovation and progress across diverse scientific disciplines beyond fluid dynamics alone.